$\Mathrm {Pin}(2) $-Monopole Floer Homology and the Rokhlin Invariant

PINp2q-MONOPOLE FLOER HOMOLOGY AND THE ROKHLIN INVARIANT

FRANCESCO LIN

Abstract. We show that the bar version of the Pinp2q-monopole Floer homology of a three- manifold Y equipped with a self-conjugate spinc structure s is determined by the triple cup product of Y together with the Rokhlin invariants of the spin structures inducing s. This is a manifestation of mod 2 index theory, and can be interpreted as a three-dimensional

counterpart of Atiyah’s classic results regarding spin structures on Riemann surfaces. x In [Lin15a] we introduced for each closed oriented three-manifold pY, sq equipped with a self-conjugate spinc structure (i.e. s “ ¯s) the Pinp2q-monopole Floer homology groups

HS ˚pY, sq, HS ˚pY, sq, HS ˚pY, sq. 3 These are graded modules over the ring R “ FrV,Qs{Q , where V and Q have degrees x respectively ´4 and ´1, and F is the ﬁeld with two elements. To deﬁne them, one exploits the 1 1 Pinp2q “ S Y j ¨ S Ă H symmetry of the Seiberg-Witten equations, and in the case b1pY q “ 0 they are the analogue of Manolescu’s invariants ([Man16]) in the context of Kronheimer and Mrowka’s monopole Floer homology ([KM07]). In particular, they can be used to provide an alternative disproof of the Triangulation conjecture. We refer the reader to [Lin16] for a friendly introduction to the construction, and to [Man13] for a survey on the Triangulation conjecture.

In the present paper, we will focus on the simplest of the three invariants, HS ˚pY, sq. This only involves reducible solutions and, heuristically, it computes the homology of the boundary of the moduli space of conﬁgurations. It is shown in [KM07] that their monopole Floer homology HM ˚pY, sq is determined entirely by the triple cup product 3 3 1 YY :Λ H pY ; Zq Ñ Z a1 ^ a2 ^ a3 ÞÑ xa1 Y a2 Y a3, rY sy.

arXiv:1708.07879v1 [math.GT] 25 Aug 2017 This is a key step in the proof of their non-vanishing theorem (Corollary 35.1.3 in [KM07]), which is in turn one of the main ingredients of Taubes’ proof of the Weinstein conjecture in dimension 3, see [Tau07]. In our set-up, recall that there is a natural map tspin structuresu Ñ tself-conjugate spinc structuresu which is surjective and has ﬁbers of cardinality 2b1pY q. This should be compared to the Bockstein sequence

1 1 2 ¨2 2 ¨ ¨ ¨ ÝÑ H pY ; Zq ÝÑ H pY ; Fq ÝÑ H pY ; Zq ÝÑ H pY ; Zq ÝÑ ¨ ¨ ¨ , which implies that the set of spin structures inducing s (which we denote by Spinpsq) is an 1 aﬃne space over H pY ; Zq b F. We will denote an element of Spinpsq by s. For each spin 1 2 FRANCESCO LIN structure s there is the (rational) Rokhlin invariant µpsq “ σpW q{8 mod 2 where W is any four-manifold whose boundary is Y and on which s extends. We will think of the collection of the Rokhlin invariants of the spin structures in Spinpsq as a map µs (called the Rokhlin map). With this in hand, we are ready to state our main theorem.

Theorem 1. The Floer homology group HS ˚pY, sq is determined by the triple cup product 3 YY and the Rokhlin map µs. More precisely, given pY0, s0q and pY1, s1q, suppose that: ‚ there exist (respectively linear and aﬃne) isomorphisms 1 1 ϕ : H pY0; Zq Ñ H pY1; Zq Φ : Spinps0q Ñ Spinps1q such that the linear part of Φ is the reduction modulo 2 of ϕ; we have ϕ˚ 3 3 , and µ Φ µ is constant. ‚ pYY1 q “ YY0 s1 ˝ ´ s0 Then there is an isomorphism of graded R-modules

HS ˚pY0, s0q ” HS ˚pY1, s1q, up to an overall grading shift.

The two main protagonists of the theorem, the Rokhlin map µs and the triple cup product 3 YY are in fact intimately connected.

Proposition 1. The Rokhlin map µs on Spinpsq is cubic, with its cubic part given by the reduction modulo 2 of the triple cup product.

We point out that while our main result shows that HS ˚pY, sq is determined by rather explicit topological data, the actual computation of the group is in general quite laborious. We will provide several concrete examples in Section 3.

The key inspiration for these results is the classic paper [Ati71]. The main difference is that while [Ati71] focuses on spin manifolds of dimension 8k ` 2 (for which the spin Dirac operator is skew-adjoint), much of the content of the present paper can be generalized to spin manifolds of dimension 8k ` 3 (for which the spin Dirac operator is quaternionic). The key ideas in the proof of our main result are the following. There is a natural involution on the torus of flat connections 1 1 T “ H pY ; iRq{2πiH pY ; Zq given by conjugation. This has exactly 2b1pY q fixed points corresponding to the spin connections of the elements of Spinpsq or, equivalently, the flat connections with holonomy t˘1u. Furthermore, the spinor bundle S Ñ Y is quaternionic (with j acting from the right), and the family of Dirac operators tDBu parametrized by rBs P T satisfies

DBpΨ ¨ jq “ pDB¯ Ψq ¨ j.

The key point is that HS ˚pY, sq can be shown to depend only on the homotopy class of this family of operators with involution, and the latter can be determined via a suitable K- theoretic index theorem. As in the case of [Ati71] (see also [AS71b]), such a homotopy class cannot be recovered purely in cohomological terms. Fortunately enough, the information lost passing to cohomology can be fully recovered in terms of the mod 2 spectral ﬂow between the PINp2q-MONOPOLE FLOER HOMOLOGY AND THE ROKHLIN INVARIANT 3

Dirac operators corresponding to the spin connections, which is in turn determined by the Rokhlin map µs.

Plan of the paper. In Section 1, after discussing a suitable K-theoretic framework, we describe the index theorem relevant to our problem. In Section 2 we discuss the specialization of this index theorem to our case of interest, and prove Theorem 1 and Proposition 1. In Section 3 we provide some explicit examples of computations. Acknowledgements. I am grateful to Tom Mrowka for many inspiring discussions and for his lectures about Atiyah’s paper during his Riemann Surfaces class in Spring 2013 at MIT. This work was supported by the the Shing-Shen Chern Membership Fund and the IAS Fund for Math.

1. Quaternionic K-theory and the index theorem for families In this section we discuss the index theorem which is relevant to our situation. While we were not able to ﬁnd in the literature the exact result we will need, the content of this section is quite standard and the proofs are straightforward adaptations of those appearing in the classical papers in the subject (to which we will refer for more details).

Quaternionic K-theory. We start by discussing a simple variant of Atiyah’s real K- theory of spaces with a involution KR introduced in [Ati66]. The key observation behind the following construction is that a real vector space can be thought as a complex vector space E equipped with a complex antilinear map squaring to 1E, while a quaternionic vector space is a complex vector space E equipped with a complex antilinear map squaring to ´1E. We will denote the latter by j. From this description we see that for any space X there are natural product maps KOipXq b KOjpXq Ñ KOi`jpXq (1) KOipXq b KSpjpXq Ñ KSpi`jpXq KSpipXq b KSpjpXq Ñ KOi`jpXq, given by tensor products.

Suppose now X is a compact Hausdorff space equipped with an involution (denoted either n by τ or x ÞÑ x¯). The key example is the n-dimensional torus pR{2πZq with the involution n induced by x ÞÑ ´x. We will denote this space with involution by T or, when the dimension is not important, simply by T.A real (resp. quaternionic) vector bundle E is a bundle 2 equipped with a complex antilinear map j covering τ such that j is 1E (resp. ´1E). The Grothendieck group of real vector bundles was introduced in [Ati66] and is denoted by KRpXq. It is important to remark that it is not the Z2-equivariant K-theory of X. Definition 1. Given a space with involution X, its quaternionic K-theory KQpXq is defined to be the Grothendieck group of quaternionic vector bundles. There are natural maps KpXq Ð KQpXq Ñ KSppXτ q 4 FRANCESCO LIN given respectively by forgetting the quaternionic structure and restricting to the fixed points of the involution. If X˚ is a space with basepoint ˚ (which we assume to be a fixed point for τ), we can define the reduced quaternionic K-theory KQpX˚q as the kernel of the augmentation map to

KQpX˚q Ñ KQĄp˚q. As usual, we can then extend the deﬁnition of KQ to deﬁne a cohomology theory on spaces with or without basepoints via

í i KQ pX˚q “ KQpΣ X˚q KQípXq “ KQpΣipX Y ˚qq. Ą Ą Here by ΣX we denote the reduced suspension ˚ Ą r0, 1s ˆ X˚{pt0u ˆ X˚ Y t1u ˆ X˚ Y r0, 1s ˆ t˚uq with the involution induced by pt, xq ÞÑ pt, τpxqq. We can also define the alternative suspension ΣX˚ with the same underlying space but involution induced by pt, xq ÞÑ p1 ´ t, τpxqq. The following is the key version of Bott periodicity for this version of K-theory. r Proposition 2. There are canonical isomorphisms KQi`8pXq – KQipXq

i j i`j KQ pΣ X˚q – KQ pX˚q ˜ Hence, roughly speaking, suspensionĄ r by Σ isĄ the inverse of suspension by Σ on reduced KQ-theory. Proof. The proof readily using analogue of the products (1) given by KQipXq b KSpjp˚q Ñ KRi`jpXq KRipXq b KSpjp˚q Ñ KQi`jpXq, and the periodicity theorems for KR and KSp. For example, using the above we have

i 1 i´8 1 KQ pT q “ KQ pT q “ i´1 7 1 i´1 6 7 i´7 i´8 “ KQ pΣ pT Y ˚qq “ KSp pS _ S q “ KSp p˚q ‘ KSp p˚q, where we used that T ^ XĄis naturally identiﬁedĆ with ΣX. Recall the basic computation for KSp-theory following from Bott periodicity r i 0 1 2 3 4 5 6 7

i KSp´ p˚q Z 0 0 0 Z Z{2Z Z{2Z 0 so that, for example 1 1 (2) KQ pT q “ Z{2Z. More in general, we have the following. PINp2q-MONOPOLE FLOER HOMOLOGY AND THE ROKHLIN INVARIANT 5

Lemma 1. We have the isomorphism

1 n an bk KQ pT q “ Z2 ‘ Z where n n a “ b “ n k n k k”1,2 mod 8 ˆ ˙ k“3,7 mod 8 ˆ ˙ 1àďkďn 1àďkďn Proof. From the discussion above, we see that 1 n 7 n KQ pT q “ KQpΣ pT qq. Recall the basic identity (see for example [Hat02], Proposition 4I.1) Ą ΣpX ˆ Y q “ ΣX _ ΣY _ ΣpX ^ Y q, n 1 n 1 which also holds at the level of space with involutions. Applying this to T “ T ˆ T ´ , by the periodicity theorem we obtain that

i n i`1 n KQ pT q “ KQ pΣT q i`1 i`1 i`1 KQ Σ 1 KQ Σ n´1 KQ Σ n´1 Ą “ Ą p T q ‘ p T q ‘ p pT ^ T qq i i i`1 KQ 1 KQ n´1 KQ n´1 . “ Ą pT q ‘ ĄpT q ‘ pTĄ q From this it inductively follows that Ą Ą n Ą 1 n k ‘pnq KQ pT q “ KQ pTq k , k“1 à and the result follows. Ą

An index theorem for quaternionic families. Consider a quaternionic bundle on a compact Riemannian manifold E Ñ M, with j acting from the left. A quaternionic family ζ is a continuous family tTpupPP of first order elliptic self-adjoint operators 8 8 Tp : C pEq Ñ C pEq where P is a space with involution p ÞÑ p¯ and we have 8 Tpps ¨ jq “ pTp¯sq ¨ j for every s P C pEq. To such an object we can associate a topological index and an analytical index as follows (see [APS76], which treats the classical case without involutions, for more details). Let π : SM Ñ M the projection from the unit cotangent bundle. For each p P P , the symbol of Tp defines ˚ a self-adjoint automorphisms σp of the bundle π E Ñ SM. This defines a decomposition ˚ ` ´ π E “ Ep ‘ Ep in positive and negative eigenspaces. In particular, we get a vector bundle ` ` Eζ “ Ep Ñ SM ˆ P pPP ď ` which is indeed quaternionic for the action of j on E, hence a class rEP s P KQpSM ˆ P q. The symbol class of ζ is defined to be the image of this class under the coboundary map δ : KQ1pSM ˆ P q Ñ KQ1pTM ˆ P q. 6 FRANCESCO LIN

More concretely, this can be described as follows (see Lemma 3.1 in [APS76]). We say that ¯ a symbol is positive/negative if Eζ “ 0. Two self-adjoint symbols are said to be stably equivalent if σ ‘ α ‘ β « σ1 ‘ α1 ‘ β1 with α, α1 positive and β, β1 negative. Then the image of the coboundary map δ is naturally in bijection with stable equivalence classes of self-adjoint symbols. Finally, we define the topological index to be the image of the symbol class under the index map KQ1pTM ˆ P q Ñ KQ1pP q, see [AS71a] for more details about the latter. Fix now a quaternionic separable Hilbert space pH, jq, and consider the space Op of complex linear Fredholm self-adjoint operators. This comes with a natural involution sending the operator T to the operator v ÞÑ ´pT pv ¨ jqq ¨ j, the fixed points of which are exactly the quaternionic linear operators. With this involution, this is naturally a classifying space for KQ1, i.e. 1 (3) KQ pP q “ rP, OpsZ{2Z where on the right-hand side we consider the homotopy classes of equivariant maps. The proof of this fact follows quite closely the non-equivariant case. First of all, one identifies the space Fred0 of Fredholm operators on H of index zero (with the same involution as above) as a classifying space for KQ (see for example [Ati67]). Then, one shows that there is an equivariant homotopy equivalence Ą (4) Op « ΩFred0 where, for a space with involution X, τ we denote its ΩX loop space (based at a fixed point p q r of τ) equipped with the involution sending a based loop γptq for t P S1 to the loop τpγpt¯qq. The operation Ω is the adjoint of Σ.˜ To show (4), fix ar quaternionic linear operator T0 P Op, and consider for T P Op the loop γT ptq in Fred0 given by r i cosptq ` sinptqT, t P r0, πs γT ptq “ #i cosptq ´ sinptqT0, t P rπ, 2πs, which we think as based at t “ 3{2π. As (right) multiplication by i is sent by conjugation to í, if we consider the involution sending t to π ´ t mod 2π (and t “ 3{2π as the basepoint) we obtain an equivariant map

Op Ñ ΩFred0, where the loop space is based at T0. This maps is essentially the one introduced in [AS69], hence it is a homotopy equivalence. r By (3), the quaternionic family of operators ζ parametrized by the space with involution P determines (after performing a standard trick to make them Fredholm) a class in KQ1pP q called the analytical index. We then have the following. Proposition 3. For a quaternionic family of operators ζ parametrized by P , the topological index coincides with the analytical index in KQ1pP q. PINp2q-MONOPOLE FLOER HOMOLOGY AND THE ROKHLIN INVARIANT 7

The proof of the result follows very closely that of the classic index theorem for families of self-adjoint operators (see [APS76], to which we refer for details). For example, using a loop construction as above one obtains a quaternionic family of elliptic symbols giving rise to an element in KQ0pS1 ˆ X ˆ TY q, and similarly the family of operators induces a class in KQ0pS1 ˆ Xq. From this, one reduces to the usual index theorem (cfr. [Ati66]).

2. Applications to monopole Floer homology We now discuss the implications of the index theorem discussed in the previous section for monopole Floer homology. As a technical remark, in the present section we will be working in the setting of unbounded self-adjoint operators S˚pH : H1q on a Hilbert space H with a suitable dense subspace H1 introduced in Chapter 33 of [KM07]. This diﬀers from that employed in the previous section (where we considered the space Op) and in the classic papers on index theory (as for example [APS76]), but the main results are readily adapted as in [KM07]. The reader not too interested about these technical aspects should think of SpH : H1q as a suitable generalization of the space of compact self-adjoint perturbations of 2 2 Dirac operators (in which case H “ L pY ; Sq and H1 “ L1pY ; Sq). Families of Dirac operators. While the results we will discuss hold more in general for p8k ` 3q-dimensional spin manifolds, we will focus on the case of of three-manifolds. Let Y be a three-manifold equipped with a self-conjugate spinc structure s, and consider the family of

Dirac operators tDBurBsPT parametrized by the b1pY q-dimensional torus T of flat connections. The torus T comes with the natural involution by conjugation (or, equivalently, x ÞÑ ´x), which has exactly 2b1pY q fixed points corresponding to the spin connections of the elements of Spinpsq. We start by discussing a key example. Suppose b1pY q “ 1. We know from (2) that 1 1 KQ pT q “ Z{2Z, so that there are exactly two homotopy classes of families of quaternionic 1 operators. These can be described explicity as follows. Given any family over T , the operators at 0 and π are quaternionic, so that in particular their kernel is even dimensional over C. This implies that the mod 2 spectral flow between the two operators is well-defined. Indeed, both cases can be realized: in the case of S2 ˆ S1 we have even spectral flow, while in the case of the zero surgery on the trefoil we have odd spectral flow (see Chapter 4 of [Lin15a]). Hence the mod 2 spectral flow is a complete invariant for families. With this in mind, we prove the following.

Proposition 4. Let tDBurBsPT be the family of Dirac operators for pY, sq. Then its index 1 in KQ pTq is determined by the triple cup product of Y together with the mod 2 spectral ﬂow between the Dirac operators corresponding to the spin connections of the elements of Spinpsq. Proof. Consider the map 1 1 (5) KQ pTq Ñ K pTq which forgets the involution. This map is injective after tensoring with Q. This is clear from the computation of the ﬁrst group (see Lemma 1), and the fact that statement is true for the natural map KSpip˚q Ñ Kip˚q. Of course, this map simply sends the family to its index (as a family of self-adjoint operator), which is in turn determined via the Chern character 1 odd ch : K pTq Ñ H pT; Qq 8 FRANCESCO LIN by the triple cup product, as it follows from the usual index theorem for families (see Lemma 1 35.1.2 of [KM07] for an explicit computation). An important point here is that K pTq is torsion-free, so no information is lost by taking the Chern character. Hence we only need to 1 understand the torsion of KQ pTq, which is according to Lemma 1 a suitable direct sum of Z{2Zs. 1 Pick a basis x1,..., xn of H pY, Zq. This determines a splitting in one-dimensional factors T “ T1 ˆ ¨ ¨ ¨ ˆ Tn, and (choosing a base spin structure s0) we can also use it to identify the set Spinpsq with subsets of t1, . . . , nu. From the computation in Lemma 1 we then know that 1 the summands Z{2Z to KQ pTq arise from wedge summands of the form

Ti1 ^ ¨ ¨ ¨ ^ Tik where k “ 1 or 2 modulo 8. Consider now a standard equivariant loop γ whose ﬁxed points are precisely s0 and the spin structure corresponding to ti1, . . . , iku. Then the suspension of the inclusion γ ãÑ T factors through the suspension of

γ ãÑ Ti1 ^ ¨ ¨ ¨ ^ Tik . The latter induces an isomorphism on KQ1: this follows for k “ 2 from the ring structure of KSp˚p˚q, and in general from the periodicity theorem (see Proposition 2). Hence the component in each Z{2Z summand can be interpreted in terms of the restriction of the family to a certain equivariant loop γ, which is in turn determined by a certain mod 2 spectral ﬂow by the discussion on the case b1 “ 1.

Coupled Morse homology. Given a family of operators L in S˚pH : H1q parametrized by a smooth manifold Q, in Chapter 33 of [KM07] the authors construct the coupled Morse ¯ homology H˚pQ, Lq, which is a relatively graded module over FrUs with U of degree ´2. We can assume in our context that the family has no spectral ﬂow around loops in Q, so that the grading is absolute. We very quickly recall its deﬁnition, and refer the reader to [KM07] for more details. Fix a metric and choose a generic Morse function f on Q. After a small perturbation of the family, we can assume that the following genericity assumption holds:

(˚) for all critical points q of f, the operator Lq has no kernel and simple spectrum. ¯ The chain complex for coupled Morse homology C˚pQ, Lq is generated over F by the pro- jectivizations of the eigenspaces of the operators at the critical points (which are all one- dimensional by (˚)). We then look at equivalence classes of pairs of paths pγptq, φptqq where γ is a Morse trajectory for f and φ is a path in the unit sphere of H satisfying a given differential equation. The differential counts the number of these paths in zero dimensional moduli spaces. The two key properties of this construction are the following: ¯ ‚ The coupled Morse homology H˚pQ, Lq only depends on the homotopy class of the 1 family L, hence on the corresponding element in K pQq “ rQ, S˚pH : H1qs. ‚ If pY, sq is a three-manifold equipped with a torsion spinc structure, the result of the construction applied to the family of Dirac operators parametrized by flat connections

tDBurBsPT is HM ˚pY, sq. An analogous construction can be performed if the manifold Q comes with an involution τ and the family of operators L is quaternionic for this involution. We will assume that the ﬁxed points of τ are isolated, so that it is locally modeled on x ÞÑ ´x. The main complication is that now the operators at the ﬁxed points of the action are quaternionic- linear, so the transversality assumption p˚q cannot be achieved (respecting the involution) as PINp2q-MONOPOLE FLOER HOMOLOGY AND THE ROKHLIN INVARIANT 9

the eigenspaces will always be even-dimensional over C. The problem can be solved, as in [Lin15a], by allowing Morse-Bott singularities of a very specific kind. Indeed, generically the operators at the fixed points of τ will have no kernel and two-dimensional eigenspaces: each of these will give rise, after projectivization, to a copy of S2 on which the involution j acts as the antipodal map. With this in mind, the construction of Pinp2q-monopole Floer from [Lin15a] carries over without significant differences (see also [Lin16] for an introduction): indeed, as we do not have to deal with boundary obstructedness phoenomena, the technical details are ¯ significantly easier in this case. The output is a version of the chain complex C˚pQ, Lq whose ¯ homology is H˚pQ, Lq which is equipped with a natural chain involution. We then define ¯ τ H˚ pQ, Lq to be the homology of the invariant subcomplex. This is naturally a module over R. We record the main features of this construction in the following result, whose proofs follows along the lines of the results cointained in [Lin15a] (and in fact much simpler).

Proposition 5. For any quaternionic family of operators L in S˚pH : H1q on pQ, τq there ¯ τ is a well-deﬁned equivariant coupled Morse homology group H˚ pQ, Lq, which is an absolutely graded module over R. The following properties hold: ¯ τ ‚ H˚ pQ, Lq only depends on the homotopy class of the quaternionic family, hence on 1 the corresponding element in KQ pQq “ rQ, S˚pH : H1qsZ{2Z; ‚ If pY, sq is a three-manifold equipped with a self-conjugate spinc structure, the result of the construction applied to the family of Dirac operators parametrized by ﬂat con-

nections tDBurBsPT is HS ˚pY, sq. Putting the pieces together, we can ﬁnally prove the main result of the present paper. Proof of Theorem 1. Using the equivariant coupled Morse homology introduced above, we see that HS ˚pY, sq only depends on the homotopy class of the quaternionic family of operators tDBurBsPT, hence by Proposition 4 only on the triple cup product on Y and the mod 2 spectral ﬂow between the Dirac operators corresponding to elements of Spinpsq. So, we only need to show that the latter are determined by the Rokhlin invariants of the spin structures. This can be seen by looking at the absolute gradings in the chain complex (as introduced in Chapter b Y 28 of [KM07]). Choose a standard equivariant Morse function f on T, so that its 2 1p q critical points correspond to the elements of Spinpsq. Given two spin structures s0 and s1 with corresponding spin connections B0 and B1, we have that the zero-dimensional chains in a stable critical submanifold Ci over pBi, 0q have relative grading

indf ps0q ´ indf ps1q ´ 2sfpDB0 ,DB1 q mod 4.

On the other hand, zero-dimensional chains in Ci have absolute grading ´σpWiq{4 ` indf psiq modulo 4, where Wi is any manifold whose boundary is Y on which si extends. For this computation we exploit the fact that on a spin four-manifold the Dirac operator is quaternionic, so that its (real) index is divisible by 4 (see also Chapter 4 of [Lin15a]). Comparing this with the formula above, we see that the mod 2 spectral ﬂow between DB0 and DB1 is exactly the diﬀerence between the Rokhlin invariants of s0 and s1.

Rokhlin invariants as a cubic map. The rest of this section is devoted to understand the structure of the Rokhlin invariants of a pair pY, sq, and in particular to prove Proposition 1. First, recall some features of spin manifolds in low-dimensions (see for example [Kir89]). Spin In dimension one, the spin cobordism group is Ω1 “ Z{2Z, the generator being the trivial double cover of the circle (see Figure 1). We will refer to this as the Lie structure, and denote 10 FRANCESCO LIN

Figure 1. The two spin structures on the circle. The one on the left corresponds to the non-trivial double cover of the circle, and extends to the disk. The one of the right is the Lie structure, it corresponds to the trivial double cover of the circle, and does not extend.

Spin it by sLie. In dimension two, Ω2 “ Z{2Z, and the spin cobordism class is determined by the Arf invariant of pΣ, sq. More precisely, by restricting to loops the spin structure s determines a map Spin q : H1pΣ, Zq b F Ñ Ω1 “ Z{2Z which is a quadratic refinement of the intersection product. The Arf invariant of this quadratic form in Z{2Z is then the spin cobordism class of pΣ, sq. Consider now the three-manifold Y “ S1 ˆ S1 ˆ S1. This has 8 spin structures, 7 with Rokhlin invariant 0 and one with Rokhlin invariant 1. All the spin structures are all products of spin structures on the circle. If at least one of the factors is not the Lie structure, then it extends to a manifold diffeomorphic to S1 ˆ S1 ˆ D2, which has signature zero. On the other hand, the product of the Lie structures (which we denote by s0 “ sLie ˆ sLie ˆ sLie) extends to a manifold with signature ´8, namely the complement of a regular fiber of the 1 elliptic fibration Ep1q Ñ CP (see [Kir89]). With this in mind, we can proceed on the proof of Proposition 1. Proof of Proposition 1. Consider two spin structures s and s1 in Spinpsq, so that they differ 1 by an element x P H pY ; Zq b F. Let Σ Ă Y be an oriented surface Poincarédual to x. We claim that the difference of their Rokhlin invariants satisfies 1 Spin µpsq ´ µps q “ rs|Σs P Ω2 “ Z{2Z. To show this, we construct a suitable spin cobordism between pY, sq and pY, s1q as follows. Consider the manifold Y ˆ r´1, 1s, with fixed spin structures s and s1 at the boundary. Then Σ ˆ t0u is a characteristic surface (in a relative sense): there is a spin structure on its complement restricting to s and s1 at the boundaries and which induces the non-trivial element Spin in Ω1 on the unit circle of a normal fiber of Σ. The proof of this is a direct generalization of the closed case, see [Kir89]. Consider now BnbhdpΣ ˆ t0uq, which is naturally identified 1 with S ˆ Σ. This has the induced spin structure sLie ˆ s|Σ. Hence to find a spin cobordism from pY, sq to pY, s1q it suffices to find a spin manifold whose boundary is S1 ˆ Σ on which this spin structure extends, and glue it in. The example of the three-torus discussed above 1 2 (which is the case S ˆ T ) readily implies that if s|Σ is trivial, one can find an extension to a manifold with σ “ 0 modulo 16, and if s|Σ is not, the one can find an extension to a PINp2q-MONOPOLE FLOER HOMOLOGY AND THE ROKHLIN INVARIANT 11

manifold with σ “ 8 modulo 16. This is because, from the properties of the Arf invariant, g 2 writing Σ “# T , s|Σ bounds if and only if the number of restriction to the summands s|T 2 that do not bound is even. To conclude the proof, we need to show that the map 1 H pY ; Zq b F Ñ Z{2Z

x ÞÑ rs|PDpxqs is cubic. To see this, ﬁx a basis x1,..., xn, with transverse dual surfaces Σ1,..., Σn. Then if x “ λixi, and Σ is dual to x, we have

ř rs|Σs “ λirs|Σi s ` λiλjrs|ΣiXΣj s ` λiλjλkrs|ΣiXΣj XΣk s, as it can be seen directlyÿ from a cutÿ and paste argumentÿ (see for example [Kir89]). Finally, the term rs|ΣiXΣj XΣk s is clearly the triple cup product xxi Y xj Y xk, rY sy modulo two. Indeed, the proof shows that not only the function is cubic (with cubic part determined by triple cup product), but also that the linear and quadratic coeﬃcients can be determined very explicitly in terms of embedded surfaces. For example, in the case of the three-torus discussed above, ﬁx s0 as the base spin structure. Using this to identify the set of spin structures with 1 H pY ; Zq b F, the map µ is 1 on any non-zero element. Denoting by xi the generator of the circle in the ith factor, we see that this is the cubic map

λixi ÞÑ 1 ` λ1 ` λ2 ` λ3 ` λ1λ2 ` λ2λ3 ` λ1λ3 ` λ1λ2λ3.

The coeﬃcientsÿ can be interpreted topologically as follows. First consider a surface Σi

Poincarédual to xi. Then s0|Σi is a product of Lie structures, hence it has Arf invariant 1, which corresponds to the coefficient of λi. Furthermore, for i ‰ j the intersection Σi X Σj Spin has induced the Lie structure, hence it is 1 P Ω1 . This corresponds to the coefficient of λiλj.

3. Examples

We now discuss some explicit computations of the group HS ˚ in terms of the topological data arising from Theorem 1. We start by recalling the simplest cases of manifolds with b1 “ 0 and 1, which were worked out in Chapter 4 of [Lin15a]. Define ´1 3 R˜ “ FrV ,V,Qs{Q , which is naturally a module over R, and ´1 ´1 I “ FrV ,V s ‘ FrV ,V sx´1y where the action of Q is an isomorphism from the first tower onto the second. Here, given a graded module M, we denote by Mxdy the module obtained by shifting the degrees up by d, i.e. Mxdyi “ Mdí. When b1 “ 0, the final result is simply the direct sum of the homologies of the critical submanifolds, hence up to a total grading shift ˜ HS ˚pY, sq “ R.

When b1 “ 1, as mentioned in the previous section, there are two cases corresponding to 1 the two elements of KQ pTq “ Z{2Z. We can pick a standard equivariant function on T with exactly two critical points. We denote by s and s1 its maximum and minimum. If the two spin structures have the same Rokhlin invariant (so that there is no mod 2 spectral flow 12 FRANCESCO LIN between the spin two Dirac operators), for each critical submanifold C over s there is a critical submanifold C1 over s1 lying in degree one less; furthermore the moduli space of trajectories 1 between C and C1 consists of two copies of CP , each mapping diffeomorphically onto the images under the evaluation map, so that ˜ 1 1 HS ˚pY, sq “ R b H pS ; Fq. In the case they have different Rokhlin invariants, for each critical submanifold C over s there is a critical submanifold C1 over s1 lying in degree one more; and the moduli space of trajectories between them consist of two points, inducing multiplication by Q2 in homology. Hence, the final result is HS ˚pY, sq “ I ‘ Ix2y. The first case is realized for example by manifolds obtained by zero surgery on a knot K Ă S3 with ArfpKq zero, while the latter happens when ArfpKq is one. With this computations in mind, we can prove a general result as follows. Here, we fix a 1 basis x1,..., xn of H pY ; Zq b F, and a base spin structure s0. Given a subset I Ă t1, . . . , nu, we denote by sI the spin structure s0 ` λixi, where λi “ 1 iff i P I. We denote by |I| the cardinality of I. ř 1 Proposition 6. There exists a spectral sequence converging to HS ˚pY, sq whose E page is

R˜x´2µpsI q ` |I|qy I ÿ so that the differential d1 has a non-zero component ˜ ˜ 1 Rx´2µpsI q ` |I|y Ñ Rx´2µpsI1 q ` |I |y 1 1 if and only I Ă I, |I | “ |I| ´ 1 and µpsI q ‰ µpsI1 q, in which case it is (up to grading shift) multiplication by Q2. Proof. To compute the Floer homology group we can proceed as in [Lin15a] and use an equivariant Morse function f : T Ñ R to perturb the equations. We can choose f to be a standard Morse function on the torus, whose critical points correspond to the 2b1pY q fixed points of the involution, hence to the set of spin structures Spinpsq. Using our basis, we can also choose the function so that the index of the critical point corresponding to sI is |I|. We can filter the Floer chain complex according to the index of the corresponding critical point of f. The E1 page is then the direct sum of the homology of the critical submanifolds, so that one obtains the first part of the statement after recalling that the index of the critical point contributes to the grading shift; the statement regarding the differentials d1 follows from our computations in the case b1 “ 1. This corollary does not provide an explicit computation for the group, but just a spectral sequence for which we know explicitly the E2-page. Indeed, also the explicit general computation of [KM07] assumes rational coefficients and exploits the formality of the de Rham cohomology of T. In our setting, unlike the case of [KM07], the differential d2 is generally non-zero, as we will see explicitly in an example.

Pinp2q-standard manifolds. We say that a three-manifold Y equipped with a self- conjugate spinc structure s is Pinp2q-standard if the triple cup product of Y vanishes and PINp2q-MONOPOLE FLOER HOMOLOGY AND THE ROKHLIN INVARIANT 13

the spin structures in Spinpsq all have the same Rokhlin invariant. We claim that in this case we have ˜ 1 HS ˚pY, sq “ R b H pT; Fq. Indeed, our main theorem implies that, up to grading shift,

b1pY q 2 1 HS ˚pY, sq “ HS ˚p# S ˆ S , s0q c where s0 is the unique torsion spin structure. The latter can be computed for example using the connected sum spectral sequence (see [Lin15b]). Indeed, we know 2 1 ˜ 1 1 HS ˚pS ˆ S , s0q “ R b H pS ; Fq and as this is free over R˜, the invariant of the connected sum is simply the tensor product over R of the invariants (as the spectral sequence collapses at the E2-page).

Manifolds with b2 “ 2. In this case the triple cup product vanishes, so that the invariant is determined by the Rokhlin invariants. If all of them coincide, then the manifold is Pinp2q- standard so that the result discussed above holds. On the other hand, the cup product of 1 two basis elements of H pY ; Zq has to be zero by Poincar´eduality. Hence the cubic form from Theorem 1 has to be linear, so that the four Rokhlin invariants coincide in pairs. In particular, in light of Theorem 1, we can compute HS ˚pY, sq as the invariant for the manifold obtained by zero surgeries on each component of a split link, one component being a trefoil and one component being unknotted. For this case 1 1 HS ˚pY, sq “ pI ‘ Ix2yq b H pS ; Fq, as it follows by looking at the connected sum spectral sequence.

Manifolds with b2 “ 3. There are several cases to discuss. First of all, if we pick a basis 1 x1, x2, x3 P H pY ; Zq, the value xx1 Y x2 Y x3, rY sy P Z is well-defined (up to sign). With a little abuse of terminology, we will refer to m as the triple cup product of Y . Recall that examples of three-manifolds with triple cup product m can be provided by the construction in [RS00] by doing surgery on a band sum of m copies of the Borromean rings, see Figure 2. First of all, we consider the case in which the triple cup product is even. By Poincaré 1 duality, the cup product on H pY ; Zq vanishes modulo two. As above, this implies that the Rokhlin function is linear. Hence either all Rokhlin invariants concide (in which case the manifold is Pinp2q-standard), or exactly half of them take one value. We can compute the 1 2 homology in the latter case as follows. We can write the torus of flat connections as T ˆ T 2 2 in such a way that the spin structures in t0u ˆ T and tπu ˆ T all have the same Rokhlin invariant. We can consider the two step filtration coming from the value in the component 1 1 T . The E page is the direct sum of the equivariant coupled Morse homologies of the families 2 2 parametrized by t0u ˆ T and tπu ˆ T , which are Pinp2q-standard. In particular, we have 1 1 2 1 2 E “ R˜ b pH pT ; Fqq ‘ pH pT ; Fqqx1y . Here the shift of the second summand` comes from the difference˘ of the Rokhlin invariants. 1 2 Furthermore, each summand H pT ; Fq has a filtration coming from the index of the Morse 2 function on T . The d1 differential maps the first summand to the second, and it also lowers 1 2 the filtration level on H pT ; Fq. Using the description of the moduli spaces in the case b1 “ 1, 14 FRANCESCO LIN the filtration preserving component is readily computed to be multiplication by Q2 (up to grading shift). Hence we have

1 2 HS ˚pY, sq “ pI ‘ Ix2yq b pH pT ; Fqq

In the case the triple cup product is odd, there are again two cases (as it can be shown by a direct inspection): either 7 spin structures attain one value and the remaining one a different one (as in the case of the three-torus), or 5 spin structures attain one value and 3 attain the other. The latter case can be realized from the general example in Figure 2 by tying a knot of Arf invariant 1 in one of the components. We already see a difference with the even case in usual monopole Floer homology: as shown in [KM07], in the odd case HM ˚pY, sq has rank 3 in each degree, rather than 4. In the first of the two possible cases, we will show that

1 3 2 3 ˜ HS ˚pY, sq “ H pT q ‘ H pT q b R. ` ˘ An analogous computation was provided in diﬀerent terms in the case of the three-torus in [Lin15a]. To see this, ﬁrst recall that the Gysin exact triangle

¨Q HS ˚pY, sq HS ˚pY, sq

HM ˚pY, sq

3 implies that (if we think of HS ˚ as a FrQs{Q -module) each cyclic summand corresponds to a i rank two subgroup F ‘ F of HM ˚). Furthermore, if this summand is isomorphic to FrQs{Q , then the generators of this subgroup diﬀer in degree by i. We look at the spectral sequence from Proposition 6. The E1 page is given by

F 3 F F F 3 3 F F F F 3 3 F F F 3 F where each column repeats 4-periodically. The groups in the ith column correspond to the critical points of index 3í. The spin structure corresponding to the last column has different Rokhlin invariant (so that it is shifted in degree by 2). We will use the convention that between two consecutive groups in the same column the map Q has the highest possible rank. 2 The differential d1 goes from one column to the one on its right, and the E page is given by PINp2q-MONOPOLE FLOER HOMOLOGY AND THE ROKHLIN INVARIANT 15

F 3 F F F 3 2 F F F F 3 3 F F 3 F

Because of the module structure, the only possible non-trivial differential d2 is the dashed one. Furthermore, as HM has rank 3 in each degree, from the discussion on the Gysin sequence above the differential d3 is forced to be the arrow drawn. Now, if d2 is not zero, the final result is

2 F F 3 2 F F F 3 3 F F 3 F

3 Of course, there are no possible extensions as FrQs{Q -modules, so that this is indeed HS ˚. 3 On the other hand, this module requires 7 generators over FrQs{Q , so that one obtains a 8 contradiction using the Gysin sequence. Hence d2 vanishes and the E -page is

3 F 3 2 F F F 3 3 F F 3 F

3 Again this requires 7 generators over FrQs{Q , but there is a non-trivial extension, as shown by the arrow, and the result follows. Finally, in the case in which exactly 5 spin structures have the same Rokhlin invariant, we have

1 3 HS ˚pY, sq “ pI ‘ Ix2yq b H pT ; Fq by an analogous argument. We just point out that this is an example in which the diﬀerential d2 of the spectral sequence is non-zero. Indeed, we can assume after a basis change that the Rokhlin map is

λ1x1 ` λ2x2 ` λ3x3 ÞÑ λ1 ` λ2 ` λ3 ` λ1λ2λ3 so that the E1-page of the spectral sequence from Proposition 6 looks like 16 FRANCESCO LIN

Figure 2. The band sum of two copies of the Borromean rings. Taking the band sum of n copies of the Borromean rings, and doing zero surgery on each of the components, one obtains a three manifold with b1 “ 3 and triple cup product n.

F 3 F F F 3 3 F F F F 3 3 F F F 3 F repeated four-periodically. So E2 is

F 2 F F F 3 F F F 3 F F F 3 F and, if we suppose that d2 is zero, we see that also d3 has to be zero (for degree reasons and because it is a map of R-modules). On the other hand, this group cannot ﬁt in the Gysin exact sequence with HM ˚.

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Department of Mathematics, Princeton University and Institute for Advanced Study E-mail address: [email protected]